Another approach of generalizing convexity and subgradients (and as a consequence critical points) is given in [1]}, [2]} where convexity with respect to a set of functions \(W\)  is defined. In such a setting \(w \in W\)  is a subgradient of \(\mathcal {F}\)  at \(x\)  whenever \(\mathcal {F}(u) \ge \mathcal {F}(x) + w(u) - w(x)\)  for any \(u \in \mathbb {X}\) . As a consequence any critical point, i.e. a point where 0 is a subgradient, will also be a global minimizer and hence such a generalization cannot be used for our purposes. In [3]} another concept of generalized gradients is discussed. In this setting the definition of the gradient depends only on neighborhoods around the point of interest. As a consequence we cannot expect the critical points to have any global properties which are necessary for the analysis in Section  hence making this generalization unfit for our analysis. However, it should be noted, that whenever convenient one might substitute any differentiability assumption on the involved functionals with Clarke's generalized gradient concept in any of the following discussions.
